Question

State Green's Theorem.

Short answer

Green's Theorem states: \[\oint_{C} (L dx + M dy) = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dA\] This theorem links the line integral over a positively oriented, piecewise smooth simple closed curve C to a double integral over the plane region D bounded by C.

Step-by-step solution

Stating the theorem in vector form

State Green's theorem as: if L and M are functions of (x, y) defined on an open region containing D and have continuous first partial derivatives there, then \[ \oint_{C} (L dx + M dy) = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dA \]

Understanding the left hand side of the theorem

The left side of the equation is a line integral over a positively oriented, piecewise smooth simple close curve C. This means we integrate the function L dx + M dy around the curve C.

Understanding the right hand side of the theorem

The right side of the equation is a double integral over the divergence of F, dA. This means we integrate the function \(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\) over the region D.

Understanding the theorem as a whole

Green's theorem states that these two integrals are equal for a region D bounded by the curve C, under the conditions stated in step 1. Hence, given a field F described by the functions L and M, we can choose to perform either the line integral or the double integral based on the convenience to find the same result.